We consider, for various values of $s$, the $n$-dimensional integral
\begin{align}
\tag{1}
W_n (s)
&:=
\int_{[0, 1]^n}
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
\end{align}
which occurs in the theory of uniform random walk integrals in the plane,
where at each step a unit-step is taken in a random direction. As such,
the integral (1) expresses the $s$-th moment of the distance
to the origin after $n$ steps.

By experimentation and some sketchy arguments we quickly conjectured and
strongly believed that, for $k$ a nonnegative integer
\begin{align}
\tag{2}
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
\end{align}
Appropriately defined, (2) also holds for negative odd integers.
The reason for (2) was long a mystery, but it will be explained
at the end of the paper.